Changing DNS when I have different email and web hosts

I've just built a new site on a NEW host at this domain www.new-domain.com

The current live site is at a different domain www.old-domain.com and different host

I want www.old-domain.com to redirect to www.new-domain.com (I have all the rewrite and redirect rules etc.)

My confusion is over what DNS records to change in order to redirect old to new AND keep email working with old domain. Here's the details:

1) I am creating a domain on my NEW webhost called old-domain.com this will be where the htaccess redirect rules are that will redirect users to new-domain.com

2) However, email is hosted separately and all the users will still be using username@old-domain.com

3) old-domain.com website is currently hosted with Rackspace, but won't be after the switch

4) new-domain.com email is hosted with Cobweb

5) old-domain.com nameservers are set to: NS.123-REG.CO.UK and NS2.123-REG.CO.UK

As I am only changing where the website of old-domain.com is hosted, I need to know what to change!

Here's my current DNS settings:

@ - MX - Priority 10 - cluster3.eu.messagel
@ - MX - Priority 20 - cluster3a.eu.message
@ - A - 1.2.3.4
@ - TXT/SPF - v=spf1 include:spf.m
autodiscover - CNAME - autodiscover.hosteds
ftp - CNAME - old-domain.
mail - A - 9.8.7.6
www - A - 1.2.3.4

Name servers: NS.123-REG.CO.UK and NS2.123-REG.CO.UK

The IP address 1.2.3.4 is with Rackspace (which has two records: @ and www). The other records I think are with Cobweb (email).

Question: If I want to point old-domain.com to the new host (Vidahost) BUT KEEP EMAIL WITH COBWEB, should I only change the 1.2.3.4 records above?

One of the reasons this came about is because 123-reg told me to change the nameservers to the new host – when the current setup has the nameservers set as 123-reg, so I thought changing this would throw email out! Ultimately, I'm only changing the website location.